A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities

Abstract

We define a `non-relativistic conformal method', based on a Schrödinger algebra with critical exponent z = 2, as the non-relativistic version of the relativistic conformal method. An important ingredient of this method is the occurrence of a complex compensating scalar field that transforms under both scale and central charge transformations. We apply this non-relativistic method to derive the curved space Newton-Cartan gravity equations of motion with twistless torsion. Moreover, we reproduce z = 2 Hořava-Lifshitz gravity by classifying all possible Schrödinger invariant scalar field theories of a complex scalar up to second order in time derivatives.

35 pages + appendices

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Cited by in corpus (20)

• Action Principle for Newtonian Gravity
• Non-Relativistic Gravity and its Coupling to Matter
• Torsional Newton-Cartan gravity from the large $c$ expansion of General Relativity
• Non-Relativistic Chern-Simons Theories and Three-Dimensional Horava-Lifshitz Gravity
• Dynamics of Carroll Strings
• Embedding Galilean and Carrollian geometries I. Gravitational waves
• Newton-Cartan Gravity and Torsion
• Oddity in nonrelativistic, strong gravity
• Three-Dimensional Extended Lifshitz, Schrödinger and Newton-Hooke Supergravity
• Carroll limit of four-dimensional gravity theories in the first order formalism
• Homogeneous Nonrelativistic Geometries as Coset Spaces
• Three-dimensional exotic Newtonian supergravity theory with cosmological constant
• Note About Hamiltonian Formalism for Newton-Cartan String and p-Brane
• Hamiltonian For String in Newton-Cartan Background
• Scale Invariance in Newton-Cartan and Hořava-Lifshitz Gravity
• Non-relativistic limit of gravity theories in the first order formalism
• Lie Algebra Expansions, Non-Relativistic Matter Multiplets and Actions
• Torsion in the Classical Spacetime Context
• A pedagogical review of gravity as a gauge theory
• Extended Bargmann FDA and non-relativistic gravity